Awesome algorithm
Public  11/28/15  7xp  Math  62.5% 
Given a vector $ A $ of rational numbers, we define the following transformation :
$ B = \{ 1\times (A_1A_2), 2\times (A_2A_3), 3\times (A_3A_4), \dots \}$
We repeat the process:
$ C = \{ 1\times (B_1B_2), 2\times (B_2B_3), 3\times (B_3B_4), \dots \}$
And so on ...
Consider now the vector
$ Z = \{ B_1, C_1, D_1, \dots \} $
If $ A = \{ \frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots \} \textrm{ find the first index such as the denominator of } Z_i \textrm{ is } \gt 10^{50} $
Answer format: index,denominator
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